Related papers: Associative triples and Yang-Baxter equation
In the continuity of our previous paper arXiv:1509.05516, we define three new algebras, $A_{n}(a,b,c)$, $B_{n}$ and $C_{n}$, that are close to the braid algebra. They allow to build solutions to the braided Yang-Baxter equation with…
In this paper we study Leavitt path algebras over quivers with relations such as quantum Yang-Baxter equation, Hecke condition, and RTT conditions. This construction allows us to produce examples of Leavitt path algebras that contain…
Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This…
We consider involutive, non-degenerate, finite set theoretic solutions of the Yang-Baxter equation. Such solutions can be always obtained using certain algebraic structures that generalize nil potent rings called braces. Our main aim here…
Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical r-matrices. In this note we explicitly quantize zero-weight super dynamical r-matrices with zero coupling constant. We also answer…
In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative)…
We consider the three-dimensional associative algebra H consisting of the 3x3 strictly upper triangular matrices whose the commutator is the Heisenberg Lie algebra. We determine the solutions of the Yang-Baxter associative equation in H.…
We study the general rational solution of the Yang-Baxter equation with the symmetry algebra sl(3). The R-matrix acting in the tensor product of two arbitrary representations of the symmetry algebra can be represented as the product of the…
We show that all strongly non-degenerate trigonometric solutions of the associative Yang-Baxter equation (AYBE) can be obtained from triple Massey products in the Fukaya category of square-tiled surfaces. Along the way, we give a…
In the first part we recall two famous sources of solutions to the Yang-Baxter equation -- R-matrices and Yetter-Drinfel$'$d (=YD) modules -- and an interpretation of the former as a particular case of the latter. We show that this result…
In this paper, we establish a bialgebra theory for Reynolds Lie algebras. First we introduce the notion of a quadratic Reynolds Lie algebra and show that it induces an isomorphism from the adjoint representation to the coadjoint…
With any even Hecke symmetry R (that is a Hecke type solution of the Yang-Baxter equation) we associate a quasitensor category. We formulate a condition on R implying that the constructed category is rigid and its commutativity isomorphisms…
We study ${\rm GL}_N$ rational $R$-matrix, which turns into the 11-vertex $R$-matrix in the $N=2$ case. First, we describe its relations to dynamical and semi-dynamical $R$-matrices using the IRF-Vertex type transformations. As a by-product…
The type-I quantum superalgebras are known to admit non-trivial one-parameter families of inequivalent finite dimensional irreps, even for generic $q$. We apply the recently developed technique to construct new solutions to the quantum…
We show that the Yang-Baxter equation is equivalent to the associativity of the algebra generated by non-commuting link operators. Starting from these link operators we build out the (FFZ) algebras, the $s\ell_q (2)$ is derived by…
We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic…
We find all non-equivalent constant solutions for classical associative Yang-Baxter equation for $gl(3)$. New examples found in the classification yield the corresponding quadratic trace Poisson brackets, double Poisson structures on free…
The BH algebra is defined by two sets of generators one of which satisfy the relations of the braid group and the other the relations of the Hecke algebra of projectors.These algebras are then combined by additional relations in a way which…
The quantum elliptic $R$-matrices of Baxter-Belavin type satisfy the associative Yang-Baxter equation in ${\rm Mat}(N,\mathbb C)^{\otimes 3}$. The latter can be considered as noncommutative analogue of the Fay identity for the scalar…
We construct nonstandard finite-dimensional representations of type C affine Hecke algebra from the viewpoint of quantum integrable models. There exists two classes of nonstandard solutions to the Yang-Baxter equation called the…